Non-decreasing and not decreasing mean very different things
In mathematics, a function that never goes down is called non-decreasing, and it is allowed to sit flat for a while. A function that is merely not decreasing could zigzag up and down wildly. That small wording gap sits at the heart of monotonic functions, a simple idea with surprisingly strong consequences.
A monotonic function either preserves order or reverses it. If larger inputs never give smaller outputs, it is monotonically increasing, also called non-decreasing; if larger inputs never give larger outputs, it is monotonically decreasing. Flat stretches are allowed, so to avoid confusion mathematicians often say weakly increasing. Demand that the output always strictly rises, or strictly falls, and you get a strictly monotone function. The idea arose in calculus and was later generalised to abstract ordered sets.
Strictness buys you something valuable: a strictly monotone function never hits the same value twice, so it can always be inverted. A weakly monotone one may plateau and then fail to be one-to-one. Sources sometimes blur this by saying monotonic when they mean strictly monotonic. A function can also be strictly monotone on part of its range and invertible there, even if not everywhere. Economists use monotonic transformation to mean relabelling by a strictly increasing function, which keeps the ranking of preferences intact; a negative one flips the order.
Monotonicity tames a function's misbehaviour. On the real line, a monotonic function has one-sided limits at every point and heads to a limit at either infinity. Its only possible breaks are jumps or removable gaps, and it can have at most countably many of them. Those breaks need not be isolated, though. By adding up small positive weights assigned to each rational number, one can build an increasing function that is continuous at exactly the irrational numbers and jumps at every rational.
The derivative side has limits too. An increasing differentiable function has a non-negative slope everywhere, but the set where things go wrong can be subtle, as the Cantor function shows. These guarantees are why monotonic functions are workhorses of technical analysis.
Source: Monotonic function
